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Integrating sine and cosine Maclaurin remainders

Published online by Cambridge University Press:  16 February 2023

Russell A. Gordon
Russell A. Gordon*
Affiliation:
Department of Mathematics and Statistics Whitman College 345 Boyer Avenue Walla Walla, WA 99362 USA e-mail: gordon@whitman.edu
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In order to state our primary results, we must first establish some notation. Let S-1 (x) = sin x and C-1 (x) = cos x, then for each non-negative integer n, let

these are the remainders of the Maclaurin series for sine and cosine, respectively. Note that for each and for each . It is known that

See [1] for several different proofs of the well-known fact that

the values of αn and βn for then follow rather easily using induction and integration by parts. (Details are provided in the Appendix.)

Type
Articles
Information
The Mathematical Gazette , Volume 107 , Issue 568 , March 2023 , pp. 96 - 102
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Copyright
© The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association

References

Jameson, G. J. O., Sine, cosine and exponential integrals, Math. Gaz. 99 (July 2015) pp. 276-289.CrossRefGoogle Scholar
Stewart, S. M., Some improper integrals involving the square of the tail of the sine and cosine functions, J. Class. Anal. 16(2) (2020) pp. 91–99.Google Scholar
Gordon, R. A., Integrating the tails of two Maclaurin series, J. Class. Anal. 18(1) (2021) pp. 83–95.CrossRefGoogle Scholar